Levi-Civita symbol

The Levi-Civita symbol, usually denoted as ε_ijk, is a notational convenience (similar to the Kronecker delta δ_ij). Its value is:

• equal to 1, if the indices are pairwise distinct and in cyclic order,
• equal to −1, if the indices are pairwise distinct but not in cyclic order, and
• equal to 0, if two of the indices are equal.

Thus

$\varepsilon _{ijk}={\begin{cases}\ \ 1&(ijk)=(123),(231),(312)\\-1&(ijk)=(132),(213),(321)\\\ \ 0&{\text{else}}\end{cases}}$

The Levi-Civita symbol changes sign whenever two of the indices are interchanged, i.e., it is antisymmetric.

Remarks:

The Levi-Civita symbol—named after the Italian mathematician and physicist Tullio Levi-Civita—mainly occurs in differential geometry and mathematical physics where it is used to define the components of the (three-dimensional) Levi-Civita (pseudo)tensor that conventionally is also denoted by ε_ijk.

The symbol has been generalized to n dimensions, denoted as ε_ijk...r and depending on n indices taking values from 1 to n. It is determined by being antisymmetric in the indices and by ε_123...n = 1.
It give rise to an n-dimensional completely antisymmetric (or alternating) pseudotensor.

The Levi-Civita symbol equals the sign of the permutation (ijk). Likewise, the generalized symbol equals the sign of the permutation (ijk...r),
(i.e., 1 for even, −1 for odd permutions and 0 if two indices are equal)
or, equivalently, the determinant of the corresponding unit vectors. Therefore the symbols are also called (Levi-Civita) permutation symbol.